Experimental set up

A compound channel setup was used for this case study. The paper including all information about the experience was found in Proust and Nikora (2020), data are available in PROUST and NIKORA (2025).

Main features for the estimation

Laboratory measurements

Take a look about all the measurements:

Experiment: `1_WSE_floodplain_realistic_uncertainty

Here the experiment with the observational data obtained in the laboratory,averaging the values of the WSE.

Calibration data

A sample was taken of all measurements shown previously:

Check MCMC

All MCMC samples:

MCMC cooked:

Corelation plot of MCMC cooked:

Check summary

Zoom into the MAP and standard deviation of the error model: WSE in mm, discharge in m3/s, velocity in m/s kmin and kmoy in m1/3/s.
a0_min a0_flood Y1_intercept Y2_intercept Y3_intercept Y4_intercept Y5_intercept
N 2001.0000000 2001.000000 2.00100e+03 2001.000000 2001.000000 2001.000000 2001.000000
Minimum 107.4760000 23.356700 1.50000e-06 5.350170 5.473510 7.743410 6.253130
Maximum 130.0730000 46.343400 1.75630e-03 17.402300 22.259300 30.763900 27.444300
Range 22.5970000 22.986700 1.75480e-03 12.052100 16.785800 23.020500 21.191200
Mean 120.2120000 32.501500 2.06800e-04 10.045000 10.231900 15.379100 15.357700
Median 119.6200000 32.874100 1.71700e-04 9.793080 10.025000 15.268400 15.098500
Q10% 114.0310000 26.350600 3.29000e-05 7.671800 7.533980 11.570400 11.315500
Q25% 116.5180000 28.121200 6.60000e-05 8.672890 8.635640 13.172900 12.937500
Q75% 124.5030000 36.109800 3.06300e-04 11.206600 11.669000 17.080600 17.506600
Q90% 126.5360000 38.714200 4.31000e-04 12.853100 13.112000 19.466900 19.593900
St.Dev. 4.8327100 4.779110 1.78100e-04 1.971020 2.200820 3.108970 3.275740
Variance 23.3551000 22.839900 0.00000e+00 3.884940 4.843600 9.665710 10.730500
CV 0.0402015 0.147042 8.61514e-01 0.196220 0.215094 0.202156 0.213297
Skewness 0.0730142 0.104323 2.22869e+00 0.635261 0.571186 0.611240 0.543355
Kurtosis -0.9667830 -0.821106 1.15867e+01 0.351366 0.620198 1.211530 0.343737
MaxPost 119.8560000 32.305200 6.37000e-05 10.483900 9.980640 13.988500 15.570500
a0_min a0_flood Y1_intercept Y2_intercept Y3_intercept Y4_intercept Y5_intercept
St.Dev. 4.83271 4.77911 0.1781380 1.97102 2.20082 3.10897 3.27574
MaxPost 119.85600 32.30520 0.0636728 10.48390 9.98064 13.98850 15.57050

Estimation of the friction coefficients

In the main channel:

Estimation of the friction coefficients

In the floodplain:

Residuals

in terms of WSE

In terms of discharge

Notes:

  • High negative corelation between a0_kmin and a0_kmoy. They can be interchangeable and the result will be the same with a weighting or factor to compensate.

Question 1:

How to reduce the corelation between a0_kmin and a0_kmoy using only a single event?

Main conclusions:

The conclusions are presented before describing the experiments, in order to highlight the key findings of the study.

In laboratory measurements:

Results of experiments:

Underlying questions:

  1. The method could identify the friction in the main channel and the floodplain using a single event in floodplain?

    Answer: 1_WSE_floodplain_realistic_uncertainty experiment, presented previously, shows calibration using a single event in floodplain lead to estimate a high number of good combinations for reproducing the observed WSE. So, it is possible but parametric uncertainty is huge due to the strong negative corelation

  2. How to reduce the corelation between kmin/kmoy ?

    Answer: Taking a pseudo-observation of the friction either in main channel or floodplain with low uncertainty to reduce the number of combinations of the parameters.

  3. A minimal number of events (2) to identify the friction in the main channel and floodplain? Should one of the events flow only in the main channel?

    Answer: This experiment must be performed in a synthetic experiment to conclude about it.

Experiment: 1_WSE_floodplain_null_uncertainty

Reduce the uncertainty in the observational data

Calibration data

A sample was taken of all measurements shown previously:

Check MCMC

All MCMC samples:

MCMC cooked:

Corelation plot of MCMC cooked:

Check summary

Zoom into the MAP and standard deviation of the error model: WSE in mm, discharge in m3/s, velocity in m/s kmin and kmoy in m1/3/s.
a0_min a0_flood Y1_intercept Y2_intercept Y3_intercept Y4_intercept Y5_intercept
N 2001.0000000 2001.000000 2.00100e+03 2001.000000 2001.000000 2001.000000 2001.000000
Minimum 107.4770000 24.728800 2.28700e-04 5.858510 5.162780 7.256530 8.535670
Maximum 128.5200000 47.267600 1.02860e-03 19.895200 19.534400 26.391600 25.783100
Range 21.0430000 22.538800 7.99900e-04 14.036700 14.371600 19.135100 17.247400
Mean 117.0700000 35.709300 4.68000e-04 10.422800 10.340100 15.183500 15.256800
Median 116.6180000 35.924800 4.40500e-04 10.168100 10.021500 15.076100 14.853100
Q10% 111.3130000 29.928700 3.08800e-04 7.961250 7.902230 11.555500 11.659700
Q25% 114.0760000 31.904100 3.64800e-04 9.012630 8.877260 13.179400 13.159800
Q75% 120.6260000 38.717500 5.48300e-04 11.564200 11.582200 17.017600 17.146600
Q90% 122.3280000 42.365800 6.64100e-04 13.277000 13.104500 18.888700 19.327900
St.Dev. 4.3978600 4.729290 1.39500e-04 2.099750 2.128930 2.922280 3.026050
Variance 19.3412000 22.366200 0.00000e+00 4.408930 4.532330 8.539750 9.156990
CV 0.0375661 0.132438 2.98147e-01 0.201457 0.205891 0.192464 0.198341
Skewness 0.1696110 0.105879 8.07301e-01 0.714238 0.737369 0.450935 0.561829
Kurtosis -0.6100620 -0.633168 4.67690e-01 0.821512 1.008740 0.474089 0.139611
MaxPost 118.9850000 33.238100 3.34500e-04 9.791470 9.998730 14.983100 14.560200
a0_min a0_flood Y1_intercept Y2_intercept Y3_intercept Y4_intercept Y5_intercept
St.Dev. 4.39786 4.72929 0.139538 2.09975 2.12893 2.92228 3.02605
MaxPost 118.98500 33.23810 0.334454 9.79147 9.99873 14.98310 14.56020

Estimation of the friction coefficients

In the main channel:

Estimation of the friction coefficients

In the floodplain:

Residuals

in terms of WSE

In terms of discharge

Notes:

  • From the first case, the uncertainty is already low, thus reducing the uncertainty in the observational data does not help to break the corelation between the parameters.
  • To assess the identifiability of the parameters, non uncertainty in observation is considered for the rest of the experiments.

Experiment: 1_WSE_floodplain_1_Kmin_null_uncertainty

Add a single pseudo-observation of the friction in the main channel with very low uncertainty

Calibration data

The WSE observation is considered without uncertainty as the previous example. Below the calibration data is shown:

event reach x t WSE Discharge Velocity Y_Kmin Y_Kmoy Yu_z Yu_Q Yu_V Yu_Kmin Yu_Kmoy
1 1 3.50 3420 0.1643105 -9999 -9999 -9999 -9999 0 -9999 -9999 -9999 -9999
1 1 5.00 3420 0.1616392 -9999 -9999 -9999 -9999 0 -9999 -9999 -9999 -9999
1 1 6.50 3420 0.1602367 -9999 -9999 100 -9999 0 -9999 -9999 0 -9999
1 1 7.00 3420 0.1603004 -9999 -9999 -9999 -9999 0 -9999 -9999 -9999 -9999
1 1 7.50 3420 0.1590004 -9999 -9999 -9999 -9999 0 -9999 -9999 -9999 -9999
1 1 9.25 3420 0.1575192 -9999 -9999 -9999 -9999 0 -9999 -9999 -9999 -9999
1 1 11.75 3420 0.1552395 -9999 -9999 -9999 -9999 0 -9999 -9999 -9999 -9999
1 1 12.50 3420 0.1543078 -9999 -9999 -9999 -9999 0 -9999 -9999 -9999 -9999
1 1 13.00 3420 0.1539768 -9999 -9999 -9999 -9999 0 -9999 -9999 -9999 -9999

Check MCMC

All MCMC samples:

MCMC cooked:

Corelation plot of MCMC cooked:

Check summary

Zoom into the MAP and standard deviation of the error model: WSE in mm, discharge in m3/s, velocity in m/s kmin and kmoy in m1/3/s.
a0_min a0_flood Y1_intercept Y2_intercept Y3_intercept Y4_intercept Y5_intercept
N 2.00100e+03 2001.0000000 2.00100e+03 2001.000000 2001.000000 2.00100e+03 2001.000000
Minimum 9.99974e+01 52.1071000 2.09000e-04 5.782940 5.750480 7.60000e-06 7.959880
Maximum 1.00002e+02 63.8039000 1.97570e-03 18.754200 19.024200 6.41560e-03 25.378100
Range 4.60000e-03 11.6968000 1.76670e-03 12.971300 13.273700 6.40800e-03 17.418200
Mean 1.00000e+02 59.3397000 4.64100e-04 10.128900 10.344600 7.17200e-04 15.158400
Median 1.00000e+02 59.3257000 4.21200e-04 9.874710 10.137300 4.06800e-04 14.991500
Q10% 1.00000e+02 57.8182000 3.25900e-04 7.646060 7.841480 1.06200e-04 11.398700
Q25% 1.00000e+02 58.6624000 3.70100e-04 8.614600 8.887270 2.06400e-04 13.211300
Q75% 1.00000e+02 60.0413000 5.21700e-04 11.302100 11.589900 8.40100e-04 16.985500
Q90% 1.00000e+02 60.8575000 6.33300e-04 12.911600 12.932400 1.66080e-03 18.978500
St.Dev. 5.14600e-04 1.2504500 1.70500e-04 2.152070 2.137930 8.37900e-04 2.945500
Variance 3.00000e-07 1.5636100 0.00000e+00 4.631390 4.570740 7.00000e-07 8.675970
CV 5.10000e-06 0.0210727 3.67430e-01 0.212468 0.206672 1.16825e+00 0.194314
Skewness 5.13807e-01 -0.6787840 3.88415e+00 0.788049 0.673140 2.61497e+00 0.368663
Kurtosis 1.27389e+01 4.1952700 2.57040e+01 0.852897 0.802783 8.41432e+00 0.186513
MaxPost 1.00000e+02 59.6315000 3.47900e-04 9.850530 10.173900 2.07500e-04 14.566800
a0_min a0_flood Y1_intercept Y2_intercept Y3_intercept Y4_intercept Y5_intercept
St.Dev. 5.146e-04 1.25045 0.170512 2.15207 2.13793 0.0008379 2.9455
MaxPost 1.000e+02 59.63150 0.347862 9.85053 10.17390 0.0002075 14.5668

Estimation of the friction coefficients

In the main channel:

Estimation of the friction coefficients

In the floodplain:

Residuals

in terms of WSE

In terms of discharge

Notes:

  • By forcing a observation very tight in the main channel and keeping a polynomial degree of 0, indicates that the estimation of the friction in the main channel is already known before the calibration. In other words, in a constant longitudinal friction, with a very tight observation of the fricion coefficient in the main channel or floodplain reduce the number of parameters to estimate of 1.
  • No more corelation is observed but this case is not realistic, as the friction coefficient in the main channel is not known accurately.
  • By fixing the friction coefficient in the main channel, the estimation of the friction coefficient in the floodplain can be better identified

Experiment: 1_WSE_floodplain_several_Kmin

Add several pseudo-observations of the friction in the main channel with a realistic uncertainty

Calibration data

event reach x t WSE Discharge Velocity Y_Kmin Y_Kmoy Yu_z Yu_Q Yu_V Yu_Kmin Yu_Kmoy
1 1 3.50 3420 0.1643105 -9999 -9999 100 -9999 0 -9999 -9999 3 -9999
1 1 5.00 3420 0.1616392 -9999 -9999 100 -9999 0 -9999 -9999 3 -9999
1 1 6.50 3420 0.1602367 -9999 -9999 100 -9999 0 -9999 -9999 3 -9999
1 1 7.00 3420 0.1603004 -9999 -9999 100 -9999 0 -9999 -9999 3 -9999
1 1 7.50 3420 0.1590004 -9999 -9999 100 -9999 0 -9999 -9999 3 -9999
1 1 9.25 3420 0.1575192 -9999 -9999 100 -9999 0 -9999 -9999 3 -9999
1 1 11.75 3420 0.1552395 -9999 -9999 100 -9999 0 -9999 -9999 3 -9999
1 1 12.50 3420 0.1543078 -9999 -9999 100 -9999 0 -9999 -9999 3 -9999
1 1 13.00 3420 0.1539768 -9999 -9999 100 -9999 0 -9999 -9999 3 -9999

Check MCMC

All MCMC samples:

MCMC cooked:

Corelation plot of MCMC cooked:

Check summary

Zoom into the MAP and standard deviation of the error model: WSE in mm, discharge in m3/s, velocity in m/s kmin and kmoy in m1/3/s.
a0_min a0_flood Y1_intercept Y2_intercept Y3_intercept Y4_intercept Y5_intercept
N 2.00100e+03 2001.0000000 2.00100e+03 2001.000000 2001.000000 2.00100e+03 2001.000000
Minimum 9.69799e+01 51.4427000 2.01100e-04 4.723730 5.544260 2.64000e-05 7.488510
Maximum 1.02902e+02 67.3106000 1.48350e-03 21.905700 18.814600 1.18611e-02 28.047800
Range 5.92210e+00 15.8679000 1.28240e-03 17.182000 13.270300 1.18347e-02 20.559300
Mean 1.00071e+02 59.1494000 4.70400e-04 10.191200 10.183600 1.57470e-03 15.265000
Median 1.00066e+02 59.1092000 4.33100e-04 9.885030 9.977880 9.85900e-04 15.013200
Q10% 9.88757e+01 56.4398000 3.04300e-04 7.597470 7.684120 2.37900e-04 11.680100
Q25% 9.94580e+01 57.9080000 3.58700e-04 8.524780 8.800370 5.04400e-04 13.165500
Q75% 1.00756e+02 60.5620000 5.35600e-04 11.540700 11.523600 1.82330e-03 17.117400
Q90% 1.01242e+02 61.8728000 6.65200e-04 13.106700 12.806200 3.70820e-03 19.204900
St.Dev. 9.74019e-01 2.2025400 1.62900e-04 2.254560 2.015270 1.70100e-03 3.113380
Variance 9.48712e-01 4.8511700 0.00000e+00 5.083040 4.061320 2.90000e-06 9.693120
CV 9.73330e-03 0.0372369 3.46397e-01 0.221227 0.197894 1.08017e+00 0.203956
Skewness 2.04720e-01 -0.1253350 1.76786e+00 0.838544 0.512311 2.23643e+00 0.556226
Kurtosis 2.70351e-01 0.7538870 5.10156e+00 1.458950 0.551952 5.63228e+00 0.690285
MaxPost 1.00066e+02 59.3612000 3.42000e-04 8.753080 9.791300 5.72200e-04 14.862800
a0_min a0_flood Y1_intercept Y2_intercept Y3_intercept Y4_intercept Y5_intercept
St.Dev. 0.974019 2.20254 0.162940 2.25456 2.01527 0.0017010 3.11338
MaxPost 100.066000 59.36120 0.341959 8.75308 9.79130 0.0005722 14.86280

Estimation of the friction coefficients

In the main channel:

Estimation of the friction coefficients

In the floodplain:

Residuals

in terms of WSE

In terms of discharge

Notes:

  • Several pseudo-observations of the friction in the main channel lead to target a set of combinations of the friction coefficients in the main channel and the floodplain leading to reproduce the observed WSE.
  • The corelation between te parameters is reduced but not completely removed.

Experiment: 1_WSE_floodplain_1_Kmoy_null_uncertainty

Similar as the second proposal but now the pseudo-observation is taken in the floodplain

Calibration data

event reach x t WSE Discharge Velocity Y_Kmin Y_Kmoy Yu_z Yu_Q Yu_V Yu_Kmin Yu_Kmoy
1 1 3.50 3420 0.1643105 -9999 -9999 -9999 -9999 0 -9999 -9999 -9999 -9999
1 1 5.00 3420 0.1616392 -9999 -9999 -9999 -9999 0 -9999 -9999 -9999 -9999
1 1 6.50 3420 0.1602367 -9999 -9999 -9999 33 0 -9999 -9999 -9999 0
1 1 7.00 3420 0.1603004 -9999 -9999 -9999 -9999 0 -9999 -9999 -9999 -9999
1 1 7.50 3420 0.1590004 -9999 -9999 -9999 -9999 0 -9999 -9999 -9999 -9999
1 1 9.25 3420 0.1575192 -9999 -9999 -9999 -9999 0 -9999 -9999 -9999 -9999
1 1 11.75 3420 0.1552395 -9999 -9999 -9999 -9999 0 -9999 -9999 -9999 -9999
1 1 12.50 3420 0.1543078 -9999 -9999 -9999 -9999 0 -9999 -9999 -9999 -9999
1 1 13.00 3420 0.1539768 -9999 -9999 -9999 -9999 0 -9999 -9999 -9999 -9999

Check MCMC

All MCMC samples:

MCMC cooked:

Corelation plot of MCMC cooked:

Check summary

Zoom into the MAP and standard deviation of the error model: WSE in mm, discharge in m3/s, velocity in m/s kmin and kmoy in m1/3/s.
a0_min a0_flood Y1_intercept Y2_intercept Y3_intercept Y4_intercept Y5_intercept
N 2.00100e+03 2.00100e+03 2.00100e+03 2001.000000 2001.000000 2001.000000 2.00100e+03
Minimum 1.15792e+02 3.29932e+01 2.08100e-04 5.669440 5.187250 7.542560 1.69000e-05
Maximum 1.22672e+02 3.30102e+01 1.33990e-03 17.042600 20.126400 30.747800 8.93260e-03
Range 6.88000e+00 1.70000e-02 1.13170e-03 11.373200 14.939200 23.205200 8.91580e-03
Mean 1.19351e+02 3.30002e+01 4.57400e-04 9.985180 10.344600 15.347600 9.48900e-04
Median 1.19345e+02 3.30000e+01 4.31900e-04 9.845290 10.207600 14.865900 6.33100e-04
Q10% 1.18555e+02 3.29997e+01 3.09100e-04 7.537140 7.828650 11.550500 1.76700e-04
Q25% 1.18880e+02 3.30000e+01 3.57000e-04 8.619260 8.932830 12.960900 3.17700e-04
Q75% 1.19792e+02 3.30004e+01 5.16200e-04 11.098300 11.668900 17.215400 1.21600e-03
Q90% 1.20155e+02 3.30008e+01 6.24800e-04 12.658800 12.998300 20.202400 2.04340e-03
St.Dev. 6.76636e-01 9.15200e-04 1.42900e-04 1.971730 2.031410 3.297080 9.98300e-04
Variance 4.57836e-01 8.00000e-07 0.00000e+00 3.887730 4.126610 10.870700 1.00000e-06
CV 5.66930e-03 2.77000e-05 3.12305e-01 0.197466 0.196373 0.214826 1.05207e+00
Skewness 1.90669e-02 1.87550e-01 1.52533e+00 0.605765 0.473508 0.812506 2.79451e+00
Kurtosis 1.30900e+00 3.11188e+01 3.75694e+00 0.461507 0.563427 1.123140 1.12049e+01
MaxPost 1.19483e+02 3.30000e+01 3.82300e-04 10.050600 8.951850 14.855300 2.04800e-04
a0_min a0_flood Y1_intercept Y2_intercept Y3_intercept Y4_intercept Y5_intercept
St.Dev. 0.676636 9.152e-04 0.142855 1.97173 2.03141 3.29708 0.0009983
MaxPost 119.483000 3.300e+01 0.382287 10.05060 8.95185 14.85530 0.0002048

Estimation of the friction coefficients

In the main channel:

Estimation of the friction coefficients

In the floodplain:

Residuals

in terms of WSE

In terms of discharge

Notes:

  • As in the second proposal, the friction coefficient in the floodplain is already known before the calibration due to the tight uncertainty in the pseudo-observation.
  • By fixing this parameters, the estimation shows that the model needs to compensate the friction in the main channel to reproduce the observed WSE.
  • The corelation between the parameters is broken but the uncertainty on the pseudo-observation is not realistic.

Bibliography:

PROUST, SEBASTIEN, and VLADIMIR I. NIKORA. 2025. “Dataset of a Laboratory Study on Flows in a Compound Open Channel with Transverse Currents.” Recherche Data Gouv. https://doi.org/10.57745/HJKRYH.
Proust, Sébastien, and Vladimir I. Nikora. 2020. “Compound Open-Channel Flows: Effects of Transverse Currents on the Flow Structure.” Journal of Fluid Mechanics 885 (February): A24. https://doi.org/10.1017/jfm.2019.973.